Towards the end of The Prestige, it's revealed that Borden accomplishes his Transported Man trick by using his twin brother as a double. It then becomes clear that he has been doing this in his personal life as well as on stage - each brother leads 'half a life', sometimes appearing as Borden and sometimes as Fallon. This is how he completes the illusion: one Borden goes into the first door, the other Borden emerges from the second door a moment later.

Angier doesn't know how Borden does the trick (and doesn't believe Borden uses a double despite being told as much by Cutter). He follows what he believes to be a clue in Borden's notebook and travels to Colorado to meet Nikola Tesla, who Angier believes can build a machine that can help him perform the trick. Angier believes that Tesla built a similar machine for Borden. However, the final pages of the notebook reveal that Tesla was merely a red herring, designed to get Angier out of Borden's way for a long time. Tesla, it turns out, had never built such a machine before.

At this point, Tesla agrees to build Angier's machine anyway. Instead of building a machine that transports matter, he builds one that duplicates it. Angier uses this machine in his act, but in order to avoid multiple Angier clones cluttering up the place, one of them has to die as part of the act.

This all makes sense, so far, but I don't buy the idea that Borden's use of the word 'Tesla' as the key to his cipher is random. It's too coincidental that he sends Angier on a wild goose chase, but Angier returns with a working machine. A film that is otherwise so tightly written shouldn't have to rely on a wild coincidence to move the narrative forward, so there must be something else going on.

Is there any evidence that Borden cloned himself using Tesla's machine?

A much larger plot device for me was actually how they achieved to find an exactly equal Angier-lookalike hanging around a bar in London, but ok.
– Napoleon Wilson♦Jul 7 '13 at 17:57

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It always struck me as odd that Angier would duplicate himself every night instead of just once and use the double!
– LiathJul 8 '13 at 7:59

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@Liath Because that was not in Angier's nature, as the movie repeatedly shows. He wasn't able to spend as much devotion to his art as Borden did. He would never had the idea of living half a life just for his show, and neither would the supposed clone (which has his own will, like the original). Just disposing them was so much easier than having them around and competing with Angier for his own life. And in the end they didn't have just the task of providing a good show, but also that of framing Borden, a task for which they had to die.
– Napoleon Wilson♦Jul 8 '13 at 8:50

5 Answers
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While this is an interresting idea (and one which others have brought up on this site before, but I'm glad someone finally turned this into a proper question), I think it contradicts both the movie's plot and its overall themes. So no, I don't think there is much evidence in favor of this theory, but there is IMHO very well evidence against it, even if not so hard an evidence (and interpretation is still a free good).

As you yourself stated in your question, Borden and his twin both lived half a life for their entire life. And in turn this is "their real trick and shows true dedication to their art". This is what actually distinguishes Borden from Angier, who always tried to go the easy way and couldn't even imagine to devote his entire life to "just a magic trick". That is why Angier doesn't get behind the old Chinese magician's trick and Borden immediately realizes it, because the old Chinese also devoted his entire non-stage life to keeping up the actual trick, like Borden did. And this is also the reason why Angier cannot believe Borden to just use a double for his Transported Man, since his own attempts with a double were not that fruitful and didn't work that well in the end, since he just wanted to do it the fast and easy way. So he thought there must be more to it than such a simple secret, but "The secret impresses noone, the trick you use it for is everything" (which might in the end even be the reason why you yourself think there to be more to Borden's trick, he achieved to deceive you, too ;-)).

So given this major theme of dedicating his whole life to his art in contrast to going the fast and easy way to just to do a good show, Borden also using a cloning machine wouldn't really fit that well. And in the end Tesla himself said he never built such a thing, and while we might not believe him, he didn't have so much reason to lie at this point (having previously claimed to have built it until Angier finds out he didn't). And even more important, what would be the reason for Borden to send Angier to Tesla if he knew he could build such a thing, since it was always planned as a distraction for Angier? While the coincidence might strike you as too coincidental, Borden sending him to the actual goal and just pretending it to be a wild goose chase would be much more of a plot-inconsistency. This also doesn't fit to Borden's reaction on this whole setup. He was as surprised as you and me that Angier actually "survived" the supposed murder, so he had absolutely no idea that Angier had a machine for cloning himself, which wouldn't be the case had he known that Tesla could construct such a device.

So that being said let's take a look at Tesla's significance as keyword and the supposed coincidence. There is still a reason for Tesla being used as keyword while coincidentally being able to build that machine, even if not as impressive a reason. In the movie he once was at a science fair or something similar in London, which both Angier and Borden attended as spectators. Borden might very well have seen Angier there and might have seen that Angier was impressed by Tesla's rather innovative science (that might have seemed like magic at that time), while all the others literally ran away in fear. So he thought Tesla might be someone that Angier could really believe to have built such a "magic" machine, which made him a reasonable keyword, given that he lived so far away.

The fact that Tesla could in the end really build such a thing was a coincidence Borden did not anticipate, and was actually an hommage to Nikola Tesla's genius, using a bit of an overfictionalized version of the actual historical person. While not being an expert in this topic I know that even the real Tesla has a reputation for being quite an innovative person who had ideas that didn't get the respect or appreciation they deserved at their time (even if a duplication machine was maybe a bit over the top even for him), which also fits to his mythical introduction in the movie and his overall impressing presence (amazingly incarnated by noone else than David Bowie). So if anyone could have built such a machine at this time it would be Tesla and Borden might have been aware of this fact (even if he underestimated him and didn't really deem it possible) and assumed that Angier thought the same. So the writer of the script (or rather the novel it was based on) payed his tribute to the real Nicola Tesla by letting his fictional version achieve true scientific magic, while presenting quite a surprising, well yeah, coincidence to the audience. It wasn't so much a coincidence more than Borden and all of us just underestimating the genius of Nikola Tesla, like real history might have done, too.

Thanks for that answer. I had forgotten about Tesla's exhibition, so it seems much less 'coincidental' now. I can believe that Angier would think that if anyone could build a cloning machine, Tesla could.
– toryanJul 8 '13 at 17:03

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I'm going to mark this as accepted. I just watched the film a second time (with girlfriend), and it all hangs together a lot better the second time around. I still think it's possible that Borden was a clone, but there's insufficient information given to substantiate that (and if you take it at face value, he definitely isn't).
– toryanJul 9 '13 at 11:37

"I still think it's possible that Borden was a clone" - That might be possible, but then definitely not with the help of Tesla. But ok, in the end everything could have been a dream... ;)
– Napoleon Wilson♦Jul 9 '13 at 15:52

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Even Angier devotes his whole life to his "trick" in the end - even more than Borden. He commits suicide every single night when he does the trick. Yes, a copy will continue to live, but the one in control and starting the trick has to kill himself. I think you are giving him a little too bad rep here.
– hisMar 6 '14 at 19:54

It should be noted that while Tesla was not responsible for Borden's trick, he did in fact make a machine for Borden, who in one scene uses a Tesla Coil on stage during a performance. So clearly Borden took a real interest in Tesla, enough to purchase one of the man's electrical devices. Thus, we have further evidence that Borden's use of Tesla as the keyword to his notebook was perhaps genuine and not merely a ruse.

Yes they were real twins. They were there from start, that is the reason Bordon was not able to answer the question when Angier asked which knot he tied to his wife, as it is possible other brother was present.

He then realizes that "Borden" was actually identical twins who shared their lives on stage and off. Just as they took turns being the man in the box and the prestige, they took turns being Borden and (by wearing makeup) Fallon. When one twin lost his fingers because of Angier, the other had his brother remove his own so they could continue to look alike in order to keep up the illusion there was only one Borden. One twin (the one alive) loved Sarah and was the girl's father, while the other twin (the one hanged) loved Olivia. So they each had half of a full life, which was enough for them but not for the women they loved.

Borden sends Angier to off trail by tricking him into wasting time with Telsa, so that Angier will not realize that they are twins. He was just trying hide his trick that they are twins. As added advantage, he sent Angier out of town chasing something of no use, getting no big competition for quite a time.

Angier then went to Tesla ultimately to be cloned, but presumably he didn't want to share his life with a clone, so he resorted to cloning himself at every performance of his illusion and killing either the clone or himself each time.

There are plot holes with both solutions. For example, if they are twins, then how come Fallon is introduced so late in their life? They would have to plan it from their childhood so how come Fallon didn't exist when they moved out on their own? Where would the other twin go during a show before Fallon? On the other hand, Tesla says he had never made a machine like the cloning machine before. He also was very against it after he made it, but why did it take him so long to realize it? If he made it for Borden, it would have existed for years.